We use the notations of Bang-Jensen and Gutin [1]} for graphs and digraphs.
By the term digraph, we always mean directed graphs with neither multi-edges nor loops.
For a (directed) graph \(G\) , \(V(G)\)  is the vertex set and \(E(G)\)  the edge (arc) set of \(G\) .
By \({\it un}(G)\)  we define the underlying undirected graph of a digraph, which can be obtained by replacing every arc \((u,v)\)  by \(\lbrace u,v\rbrace \)  and deleting multi-edges. For a subset \(V^{\prime }\subseteq V(G)\)  of a digraph \(G\)  let \(G[V^{\prime }]\)  be the subgraph induced by the vertex set \(V^{\prime }\) .
The neighborhood of a vertex \(v\)  in an undirected graph \(G\)  is \(N_G(v)=\lbrace u \mid \lbrace u,v\rbrace  \in E(G)\rbrace \) . If the graph is clear from the context, we simply say \(N(v)\) . For the directed neighborhood we distinguish between the incoming-neighbors \(N^-(v)=\lbrace u \mid (u,v)\in E(G)\rbrace \)  and the outgoing-neighbors \(N^+(v)=\lbrace u \mid (v,u)\in E(G)\rbrace \) .
For a digraph \(G\) , a strongly connected subdigraph is an induced subdigraph \(H\)  of \(G\)  such that for all vertices \(u,v \in V(H)\)  there is a directed walk from \(u\)  to \(v\)  in \(H\)  and a directed walk from \(v\)  to \(u\)  in \(H\) .
A strong component of \(G\)  is a maximal strongly connected subdigraph of \(G\) , i.e. a strong component \(H\)  of \(G\)  such that there is no vertex \(v \in V(G) \setminus V(H)\)  such that the induced subdigraph of \(G\)  generated by \(V(H) \cup \lbrace v\rbrace \)  is a strong component of \(G\) .
A digraph \(G\)  is weakly connected if \({\it un}(G)\)  is connected.
A biorientation of an undirected graph \(G\)  is obtained by replacing every edge \(\lbrace u,v\rbrace \in E(G)\)  by one or both of the arcs \((u,v)\)  and \((v,u)\) .
We call vertex \(v\)  in digraph \(G\)  a bioriented leaf if there exist \((u,v),(v,u)\in E(G)\)  and if \(v\)  is a leaf in \({\it un}(G)\) , which means that in \(un(G)\)  it holds that \(\mid N_{un(G)}(v)\mid = 1\) .
